Problem 51 - Calculus with Concepts in Calculus

Question

Graph each pair of functions. Shade the region(s) the graphs enclose. $$ f(x)=x^{2}, g(x)=x $$

Step-by-Step Solution

Answer

Shade the region between the curves from \( x = 0 \) to \( x = 1 \).

Step 1: Sketch the Graph of f(x)

The function \( f(x) = x^2 \) is a standard parabola that opens upwards with its vertex at the origin (0,0). It is symmetric about the y-axis and passes through points like (1,1), (2,4), (-1,1), and (-2,4). Draw this curve on the graph.

Step 2: Sketch the Graph of g(x)

The function \( g(x) = x \) is a linear function that passes through the origin with a slope of 1. This means it forms a straight line making a 45-degree angle with the x-axis. This line passes through points like (1,1), (2,2), (-1,-1), and (-2,-2). Draw this line on the same graph.

Step 3: Identify Intersection Points

To find the points where the graphs of \( f(x) \) and \( g(x) \) intersect, set \( x^2 = x \) and solve for \( x \). This gives us \( x^2 - x = 0 \). Factor this to obtain \( x(x-1) = 0 \), giving solutions \( x = 0 \) and \( x = 1 \). These points of intersection are (0,0) and (1,1).

Step 4: Plot the Intersection Points on the Graph

Mark the points (0,0) and (1,1) on your sketch. These points are where the graphs of \( f(x) \) and \( g(x) \) intersect.

Step 5: Determine the Shaded Region

The functions \( f(x) \) and \( g(x) \) enclose a region between these intersection points. For \( 0 \leq x \leq 1 \), \( g(x) = x \) is above \( f(x) = x^2 \). Shade the region between these curves from \( x = 0 \) to \( x = 1 \).

Key Concepts

Parabolic Functions

A parabolic function is a mathematical function that forms a curve known as a parabola. One of the simplest parabolic functions is given by the formula \( f(x) = x^2 \). This function represents a U-shaped curve on the graph, and here are some key features:

• The vertex of the parabola \( f(x) = x^2 \) is at the origin, which is the point (0,0).
• The parabola is symmetric about the y-axis, meaning it looks the same on both sides of this axis.
• It opens upwards, indicating that the arms of the parabola extend in the upward direction.
• Common points through which this parabola passes include: (1,1), (2,4), (-1,1), and (-2,4).

This function is foundational in learning about parabolas because of its simplicity and symmetry. When graphing, it's essential to mark its vertex and a few key points to accurately sketch the curve.

Linear Functions

Linear functions are among the most straightforward functions you'll encounter. They represent straight lines on a graph and are defined by an equation of the form \( y = mx + b \). In the given exercise, we have \( g(x) = x \), which is a special linear function. Here's why:

• The line \( g(x) = x \) passes through the origin (0,0), meaning it starts at this point.
• The slope \( m \) here is 1, indicating that for every unit increase in \( x \), \( y \) also increases by 1, creating a 45-degree angle with the x-axis.
• This line is often called the identity line because every value of \( x \) gives the same value in \( y \).
• It passes through points such as (1,1), (2,2), (-1,-1), and (-2,-2).

When graphing linear functions like \( g(x) = x \), simply plot several points that satisfy the equation and then draw a straight line through them.

Intersection Points

Intersection points are where two or more graphs cross each other on a graph. For the functions \( f(x) = x^2 \) and \( g(x) = x \), these intersections are points of particular interest.
To find intersection points:

• Set the equations equal to each other: \( x^2 = x \).
• Solve for \( x \): \( x^2 - x = 0 \).
• Factor the equation: \( x(x-1) = 0 \). So, the solutions are \( x = 0 \) and \( x = 1 \).

These results indicate that the graphs intersect at points (0,0) and (1,1). Intersection points tell us exactly where the behaviors of the two functions coincide, making these crucial in identifying areas that may be enclosed between curves.

Shading Regions

Shading regions in a graph helps us visualize the area that is enclosed between functional curves. For the functions \( f(x) = x^2 \) and \( g(x) = x \) in our exercise, these intersect at points (0,0) and (1,1).
The steps for identifying and shading these regions are:

• Determine which function is "above" the other between the intersection points. For \( 0 \leq x \leq 1 \), the line \( g(x) = x \) lies above the curve \( f(x) = x^2 \).
• Mark the area between these curves on the graph from \( x = 0 \) to \( x = 1 \).
• Shade this region to signify the bounded area, illustrating which part of the graph is of interest.

Understanding which regions to shade is important in calculus. It helps to set up integrals for calculating the area between curves, a critical skill in many applications of mathematics.